The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.Research supported in part by DOE CHAMMP and HPCC programs.Research partially supported by the Department of Energy, the Office of Naval Research and the Fields Institute for Research in the Mathematical Sciences.     

Characterisation of valves as sound sources: Fluid-borne sound

The sound pressure level in receiving rooms, caused by taps at the ends of pipe systems, is considered. The structure-borne sound power, from the pipes to the supporting wall, was obtained from intensity measurement of the fluid-borne sound power of the tap. The fluid-borne sound power is combined with a ratio of structure-borne sound power to fluid-borne sound power, obtained from laboratory measurements of similar pipe assemblies. Alternatively, a reception plate method is proposed, which avoids the necessity for intensity measurements. The structure-borne power into walls, to which the pipe work is attached, provides input to the standard building propagation model, which yields the predicted sound pressure level in the adjacent room.     

Second order asymptotics for the propagation speed of interfaces in the Allen-Cahn phase field model for elastic solids

For a phase field model, which consists of the elasticity equations coupled to the Allen-Cahn equation, we state an asymptotic expansion for the propagation speed of the diffusive interface. The error of the expansion is of order η², where η is the width of the interface. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rotor Stator Interaction with Many Degrees of Freedom

A many-degrees-of-freedom model of rotor stator contact is presented. The stability analysis of the synchronous motion of the simple JEFFCOTT rotor contacting a single-mass stator is extended to systems with many degrees of freedom. The stationary synchronous motion as well as its stability are validated by direct numerical integration of the differential equations. The resulting dynamics is exemplified in two examples, which may be quite different than the dynamics of the simple JEFFCOTT rotor. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Renormalization group analysis of a confined, interacting Bose gas

The renormalization group is not only a powerful method for describing universal properties of phase transitions, but it is also useful for evaluating non-universal thermodynamic properties beyond mean-field theory. In this contribution we concentrate on these latter aspects of the renormalization group approach. We introduce its main underlying ideas in the familiar context of the ideal Bose gas and then apply them to the case of an interacting, confined Bose gas within the framework of the random phase approximation. We model confinement by periodic boundary conditions and demonstrate how confinement modifies the flow equations of the renormalization group, thus changing the thermodynamic properties of the gas. Received: 20 July 2001 / Revised version: 20 August 2001 / Published online: 23 November 2001     

Evolving Microstructure and Homogenization

In this article we formulate a mathematical model for the temporally evolving microstructure generated by phase changes and study the homogenization of this model. The investigations are partially formal, since we do not prove existence or convergence of solutions of the microstructure model to solutions of the homogenized problem. To model the microstructure, the sharp interface approach is used. The evolution of the interface is governed by an everywhere defined distribution partial differential equation for the characteristic function of one of the phases. This avoids the disadvantage commonly associated with this approach of an evolution equation only defined on the interface. To derive the homogenized problem, a family of solutions of the microstructure problem depending on the fast variable is introduced. The homogenized problem obtained contains a history functional, which is defined by the solution of an initial-boundary value problem in the representative volume element. In the special case of a temporally fixed microstructure the homogenized problem is reduced to an evolution equation to a monotone operator. Received March 15, 2000